Hankel Determinants of Some Sequences of Polynomials

Ehrenborg gave a combinatorial proof of Radoux’s theorem which states that the determinant of the (n + 1)× (n + 1) dimensional Hankel matrix of exponential polynomials is x ∏n i=0 i!. This proof also shows the result that the (n + 1) × (n + 1) Hankel matrix of factorial numbers is ∏n k=1(k!) . We observe that two polynomial generalizations of factorial numbers also have interesting determinant values for Hankel matrices. A polynomial generalization of the determinant of the Hankel matrix with entries being fixed-point free involutions on the set [2n] is given next. We also give a bivariate non-crossing analogue of a theorem of Cigler about the determinant of a similar Hankel matrix.